biplotEZ

• Data matrix in CA() is different from PCA() and CVA().

• \(\mathbf{X}:r\times c\), represents the cross-tabulations of two categorical variables.

• The elements of the data matrix represent the frequency of the co-occurrence of two particular levels of the two variables.

Consider the HairEyeColor data set in R, which summarises the hair and eye color of male and female statistics students. For the purpose of this example only the male students will be considered:

cross_tab <- HairEyeColor[,,2]
cross_tab

#        Eye
# Hair    Brown Blue Hazel Green
#   Black    36    9     5     2
#   Brown    66   34    29    14
#   Red      16    7     7     7
#   Blond     4   64     5     8

• biplot() |>
  • CA() |>
  • interpolate() |> fit.measures() |>
  • samples() |> newsamples() |>
  • legend.type() |>
• plot()

Take note of the warning message:

biplot(HairEyeColor[,,2], center = TRUE) |> CA()

# Warning in CA.biplot(biplot(HairEyeColor[, , 2], center = TRUE)): Centering was
# not performed. Set biplot(center = FALSE) when performing CA().

# Object of class CA, based on 4 row levels and 4 column levels.

• It is typical to express the frequencies in terms of proportions / probabilities.

• Consider the correspondence matrix \(\mathbf{P}\):

P_mat <- cross_tab/sum(cross_tab)
P_mat

#        Eye
# Hair          Brown        Blue       Hazel       Green
#   Black 0.115015974 0.028753994 0.015974441 0.006389776
#   Brown 0.210862620 0.108626198 0.092651757 0.044728435
#   Red   0.051118211 0.022364217 0.022364217 0.022364217
#   Blond 0.012779553 0.204472843 0.015974441 0.025559105

• Row profiles (diagonal matrix) - \(\mathbf{D_r}\): update with biplotEZ

ca.out <- biplot(HairEyeColor[,,2], center = FALSE) |> CA()
ca.out$Dr

#           [,1]     [,2]      [,3]      [,4]
# [1,] 0.1661342 0.000000 0.0000000 0.0000000
# [2,] 0.0000000 0.456869 0.0000000 0.0000000
# [3,] 0.0000000 0.000000 0.1182109 0.0000000
# [4,] 0.0000000 0.000000 0.0000000 0.2587859

• Column profiles (diagonal matrix) - \(\mathbf{D_c}\):

ca.out <- biplot(HairEyeColor[,,2], center = FALSE) |> CA()
ca.out$Dc

#           [,1]      [,2]      [,3]       [,4]
# [1,] 0.3897764 0.0000000 0.0000000 0.00000000
# [2,] 0.0000000 0.3642173 0.0000000 0.00000000
# [3,] 0.0000000 0.0000000 0.1469649 0.00000000
# [4,] 0.0000000 0.0000000 0.0000000 0.09904153

• Consider the independence model:

\[\chi^2 = \frac{(\text{Observed freq.}-\text{Expected freq.})^2}{\text{Expected freq.}}\]

• Standardised Pearson residuals (\(\mathbf{S}\)):

\[ \mathbf{S} = \mathbf{D_r}^{-\frac{1}{2}}(\mathbf{P}-\mathbf{rc'})\mathbf{D_c}^{-\frac{1}{2}}\]

• In terms of the weighted row and column profiles (\(\mathbf{D_r}^{-\frac{1}{2}}\) and \(\mathbf{D_c}^{-\frac{1}{2}}\)).

• The expected frequencies represented by the product of the row and column profiles ().

• Biplot coordinates: singular value decomposition of \(\mathbf{S}\).

\[ \text{svd}(\mathbf{S}) = \mathbf{U\Lambda V'}\]

• Variant refers to the contribution of the singular values (\(\Lambda\)) in the biplot solution.

  • Row principal coordinate biplot (default):

  \[\begin{aligned} \text{Row coordinates: } \hspace{0.5 cm}&\mathbf{U\Lambda}\\ \text{Column coordinates: }\hspace{0.5 cm}& \mathbf{V}\end{aligned}\]

  • Row standard coordinate biplot:

    \[\begin{aligned} \text{Row coordinates: } \hspace{0.5 cm}&\mathbf{U}\\ \text{Column coordinates: }\hspace{0.5 cm}& \mathbf{V\Lambda}\end{aligned}\]

  • Symmetric Correspondence Analysis map:

    \[\begin{aligned} \text{Row coordinates: } \hspace{0.5 cm}&\mathbf{U\Lambda^{\frac{1}{2}}}\\ \text{Column coordinates: }\hspace{0.5 cm}& \mathbf{V\Lambda^{\frac{1}{2}}}\end{aligned}\]